Math / Geometry

Questionاوجد كل المثلثات الفيثاغورية البدانية التي طول احد الساقين فيها يساوي 80 .

Studdy Solution

STEP 1

1. A Pythagorean triple consists of three positive integers $(a, b, c)$ such that $a^2 + b^2 = c^2$ .
2. One of the legs of the triangle is given as 80.
3. We need to find all possible integer values for the other leg and the hypotenuse.

STEP 2

1. Identify the known leg and set up the equation.
2. Use the Pythagorean theorem to express the relationship between the sides.
3. Solve for possible integer values of the other leg and hypotenuse.
4. Verify and list all valid Pythagorean triples.

STEP 3

Identify the known leg of the triangle. Let $a = 80$ , where $a$ is one of the legs of the Pythagorean triple.

STEP 4

Use the Pythagorean theorem $a^2 + b^2 = c^2$ to express the relationship between the sides. Substitute $a = 80$ :
$80^2 + b^2 = c^2$

STEP 5

Calculate $80^2$ :
$80^2 = 6400$
Substitute back into the equation:
$6400 + b^2 = c^2$

STEP 6

Rearrange the equation to solve for $c$ :
$c^2 = 6400 + b^2$
$c = \sqrt{6400 + b^2}$

STEP 7

To find integer solutions, $c$ must also be an integer. Therefore, $6400 + b^2$ must be a perfect square. Iterate over possible integer values of $b$ to find such cases.

STEP 8

Start with $b = 1$ and incrementally increase $b$ to find values that make $c$ an integer.
For each $b$ , check if $6400 + b^2$ is a perfect square.

STEP 9

Continue this process until all possible integer values of $b$ are exhausted, ensuring $c$ remains an integer.

STEP 10

Verify each solution by checking if $a^2 + b^2 = c^2$ holds true for each set of values $(a, b, c)$ .
List all valid Pythagorean triples.
The valid Pythagorean triples with one leg equal to 80 are:
1. $(80, 60, 100)$
2. $(80, 84, 116)$

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Questionاوجد كل المثلثات الفيثاغورية البدانية التي طول احد الساقين فيها يساوي 80 .

Studdy Solution

STEP 1

1. A Pythagorean triple consists of three positive integers $(a, b, c)$ such that $a^2 + b^2 = c^2$ .
2. One of the legs of the triangle is given as 80.
3. We need to find all possible integer values for the other leg and the hypotenuse.

STEP 2

1. Identify the known leg and set up the equation.
2. Use the Pythagorean theorem to express the relationship between the sides.
3. Solve for possible integer values of the other leg and hypotenuse.
4. Verify and list all valid Pythagorean triples.

STEP 3

Identify the known leg of the triangle. Let $a = 80$ , where $a$ is one of the legs of the Pythagorean triple.

STEP 4

Use the Pythagorean theorem $a^2 + b^2 = c^2$ to express the relationship between the sides. Substitute $a = 80$ :
$80^2 + b^2 = c^2$

STEP 5

Calculate $80^2$ :
$80^2 = 6400$
Substitute back into the equation:
$6400 + b^2 = c^2$

STEP 6

Rearrange the equation to solve for $c$ :
$c^2 = 6400 + b^2$
$c = \sqrt{6400 + b^2}$

STEP 7

To find integer solutions, $c$ must also be an integer. Therefore, $6400 + b^2$ must be a perfect square. Iterate over possible integer values of $b$ to find such cases.

STEP 8

Start with $b = 1$ and incrementally increase $b$ to find values that make $c$ an integer.
For each $b$ , check if $6400 + b^2$ is a perfect square.

STEP 9

Continue this process until all possible integer values of $b$ are exhausted, ensuring $c$ remains an integer.

STEP 10

Verify each solution by checking if $a^2 + b^2 = c^2$ holds true for each set of values $(a, b, c)$ .
List all valid Pythagorean triples.
The valid Pythagorean triples with one leg equal to 80 are:
1. $(80, 60, 100)$
2. $(80, 84, 116)$

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Questionاوجد كل المثلثات الفيثاغورية البدانية التي طول احد الساقين فيها يساوي 80 .

Studdy Solution

STEP 1

1. A Pythagorean triple consists of three positive integers (a,b,c) (a, b, c) (a,b,c) such that a2+b2=c2 a^2 + b^2 = c^2 a2+b2=c2.2. One of the legs of the triangle is given as 80.3. We need to find all possible integer values for the other leg and the hypotenuse.

STEP 2

1. Identify the known leg and set up the equation.2. Use the Pythagorean theorem to express the relationship between the sides.3. Solve for possible integer values of the other leg and hypotenuse.4. Verify and list all valid Pythagorean triples.

STEP 3

Identify the known leg of the triangle. Let a=80 a = 80 a=80, where a a a is one of the legs of the Pythagorean triple.

STEP 4

Use the Pythagorean theorem a2+b2=c2 a^2 + b^2 = c^2 a2+b2=c2 to express the relationship between the sides. Substitute a=80 a = 80 a=80:802+b2=c2 80^2 + b^2 = c^2 802+b2=c2

STEP 5

Calculate 802 80^2 802:802=6400 80^2 = 6400 802=6400Substitute back into the equation:6400+b2=c2 6400 + b^2 = c^2 6400+b2=c2

STEP 6

Rearrange the equation to solve for c c c:c2=6400+b2 c^2 = 6400 + b^2 c2=6400+b2c=6400+b2 c = \sqrt{6400 + b^2} c=6400+b2​

STEP 7

To find integer solutions, c c c must also be an integer. Therefore, 6400+b2 6400 + b^2 6400+b2 must be a perfect square. Iterate over possible integer values of b b b to find such cases.

STEP 8

Start with b=1 b = 1 b=1 and incrementally increase b b b to find values that make c c c an integer.For each b b b, check if 6400+b2 6400 + b^2 6400+b2 is a perfect square.

STEP 9

Continue this process until all possible integer values of b b b are exhausted, ensuring c c c remains an integer.

STEP 10

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Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. A Pythagorean triple consists of three positive integers $(a, b, c)$ such that $a^2 + b^2 = c^2$ .
2. One of the legs of the triangle is given as 80.
3. We need to find all possible integer values for the other leg and the hypotenuse.

1. Identify the known leg and set up the equation.
2. Use the Pythagorean theorem to express the relationship between the sides.
3. Solve for possible integer values of the other leg and hypotenuse.
4. Verify and list all valid Pythagorean triples.

Identify the known leg of the triangle. Let $a = 80$ , where $a$ is one of the legs of the Pythagorean triple.

Use the Pythagorean theorem $a^2 + b^2 = c^2$ to express the relationship between the sides. Substitute $a = 80$ :
$80^2 + b^2 = c^2$

Calculate $80^2$ :
$80^2 = 6400$
Substitute back into the equation:
$6400 + b^2 = c^2$

Rearrange the equation to solve for $c$ :
$c^2 = 6400 + b^2$
$c = \sqrt{6400 + b^2}$

To find integer solutions, $c$ must also be an integer. Therefore, $6400 + b^2$ must be a perfect square. Iterate over possible integer values of $b$ to find such cases.

Start with $b = 1$ and incrementally increase $b$ to find values that make $c$ an integer.
For each $b$ , check if $6400 + b^2$ is a perfect square.

Continue this process until all possible integer values of $b$ are exhausted, ensuring $c$ remains an integer.